Galileo's Paradox
in [Math]
|
From: Jesse F. Hughes on 26 Nov 2006 18:35 Six Letters writes: >>The one-to-one correspondence idea is nice because it works for any >>two sets. The idea you are looking at only works if one set >>is a subset of the other. > > Yes, to set up the paradox we need to compare two sets for > which there is a 1:1C and one is a subset of the other. It isn't a > question of what works. It's a question of how the paradox is to be > resolved. But the resolution needs to also discuss set sizes for non-comparable sets. Otherwise, it's a pretty dull notion of set size. We already have the subset operation. -- Jesse F. Hughes "I want to really eat myself, so then I'll be a coalgebra." -- Quincy P. Hughes, Age 3 1/2
From: stephen on 26 Nov 2006 21:21 Six wrote: > On Fri, 24 Nov 2006 18:26:37 +0000 (UTC), stephen(a)nomail.com wrote: >>Six wrote: >>> On Fri, 24 Nov 2006 16:04:12 +0000 (UTC), stephen(a)nomail.com wrote: >> >>>>Six wrote: >>>> >>>><snip> >>>> >>>>> I want to suggest there are only two sensible ways to resolve the >>>>> paradox: >>>> >>>>> 1) So- called denumerable sets may be of different size. >>>> >>>>> 2) It makes no sense to compare infinite sets for size, neither to say one >>>>> is bigger than the other, nor to say one is the same size as another. The >>>>> infinite is just infinite. >>>> >>>>> >>>>> My line of thought is that the 1:1C is a sacred cow. That there is >>>>> no extension from the finite case. >>>> >>>>What do you mean by that? The one-to-one correspondence works >>>>perfectly in the finite case. That is the entire idea behind >>>>counting. Given any two finite sets, such as { q, x, z, r} and >>>>{ #, %, * @ }, there exists a one-to-one correspondence between >>>>them if and only if they have the same number of elements. >>>>This is the idea that let humans count sheep using rocks long >>>>before they had names for the numbers. >> >>> I love this quaint, homely picture of the origin of arithmetic. I >>> am sure that evolutionary arithmetic will soon be taught in universities, >>> if it is not already. Disregarding the anthropology, however, you have said >>> absolutely nothing about whether !:!C is adequate for the infinite case. >> >>I was addressing your claim that there was "no extension from the >>finite case". In the finite case, two sets have the same number >>of elements if and only if there exists a one to one correspondence >>between them. This very simple idea has been extended to the >>infinite case. > OK. The idea of a 1:1 corresondence is indeed a simple idea. The idea of > infinity is not. That depends on what 'idea of infinity' of you are talking about. The mathematical definition of 'infinite' is as simple as the idea of a 1:1 correspondence. There is no point in dragging philosophical baggage into a mathematical discussion. >>>>> If we want to compare the two sets for size we would write, not the >>>>> above, but: >>>> >>>>> 1 2 3 4 5 6 7 8 9 ............... >>>>> 1 2 3 4 5 6 7 8 9................ >>>>> ^ ^ ^ >>>> >>>>Why would we write that? The second line seems to have nothing >>>>to do with the second set. Why include elements that are not >>>>in the set? >> >>> It's just the awkwardness of plain text. The members of the sqaures >>> set are meant to be circled, if you like, the missing intergers being >>> supplied for comparison. >> >>But why should you bother listing the missing integers? If you >>are comparing two sets, including elements that are not in one >>of the sets in the comparision seems strange to me. If >>I am comparing the sets {Bob, Dave, George} and {Eileen, Barb, Jill} >>should I write out >> >> Adam, Bob, Chuck, Dave, Edgar, Fred, George, Hank, Igor, Jack >> ^^^ ^^^^ ^^^^^^ if you insist it matches some vague unspecified notion of "same size". It may seem unintuitive, >> >> ^^^^ ^^^^^^ ^^^^ >> >>?? How do I know what the missing elements are? >> >>The one-to-one correspondence idea is nice because it works for any >>two sets. The idea you are looking at only works if one set >>is a subset of the other. > Yes, to set up the paradox we need to compare two sets for which > there is a 1:1C and one is a subset of the other. It isn't a question of > what works. It's a question of how the paradox is to be resolved. > Thanks, Six Letters There is no need to resolve the paradox. There exists a one-to-correspondence between the natural numbers and the perfect squares. The perfect squares are also a proper subset of the natural numbers. This is not a contradiction. Stephen
From: Virgil on 26 Nov 2006 21:30 In article <ekdi3d$i15$1(a)news.msu.edu>, stephen(a)nomail.com wrote: > Six wrote: > > On Fri, 24 Nov 2006 18:26:37 +0000 (UTC), stephen(a)nomail.com wrote: > > >>Six wrote: > >>> On Fri, 24 Nov 2006 16:04:12 +0000 (UTC), stephen(a)nomail.com wrote: > >> > >>>>Six wrote: > >>>> > >>>><snip> > >>>> > >>>>> I want to suggest there are only two sensible ways to resolve the > >>>>> paradox: > >>>> > >>>>> 1) So- called denumerable sets may be of different size. > >>>> > >>>>> 2) It makes no sense to compare infinite sets for size, neither to say > >>>>> one > >>>>> is bigger than the other, nor to say one is the same size as another. > >>>>> The > >>>>> infinite is just infinite. > >>>> > >>>>> > >>>>> My line of thought is that the 1:1C is a sacred cow. That there is > >>>>> no extension from the finite case. > >>>> > >>>>What do you mean by that? The one-to-one correspondence works > >>>>perfectly in the finite case. That is the entire idea behind > >>>>counting. Given any two finite sets, such as { q, x, z, r} and > >>>>{ #, %, * @ }, there exists a one-to-one correspondence between > >>>>them if and only if they have the same number of elements. > >>>>This is the idea that let humans count sheep using rocks long > >>>>before they had names for the numbers. > >> > >>> I love this quaint, homely picture of the origin of arithmetic. I > >>> am sure that evolutionary arithmetic will soon be taught in universities, > >>> if it is not already. Disregarding the anthropology, however, you have > >>> said > >>> absolutely nothing about whether !:!C is adequate for the infinite case. > >> > >>I was addressing your claim that there was "no extension from the > >>finite case". In the finite case, two sets have the same number > >>of elements if and only if there exists a one to one correspondence > >>between them. This very simple idea has been extended to the > >>infinite case. > > > OK. The idea of a 1:1 corresondence is indeed a simple idea. The idea of > > infinity is not. > > That depends on what 'idea of infinity' of you are talking about. > The mathematical definition of 'infinite' is as simple as the > idea of a 1:1 correspondence. There is no point in dragging > philosophical baggage into a mathematical discussion. > > >>>>> If we want to compare the two sets for size we would write, not the > >>>>> above, but: > >>>> > >>>>> 1 2 3 4 5 6 7 8 9 ............... > >>>>> 1 2 3 4 5 6 7 8 9................ > >>>>> ^ ^ ^ > >>>> > >>>>Why would we write that? The second line seems to have nothing > >>>>to do with the second set. Why include elements that are not > >>>>in the set? > >> > >>> It's just the awkwardness of plain text. The members of the sqaures > >>> set are meant to be circled, if you like, the missing intergers being > >>> supplied for comparison. > >> > >>But why should you bother listing the missing integers? If you > >>are comparing two sets, including elements that are not in one > >>of the sets in the comparision seems strange to me. If > >>I am comparing the sets {Bob, Dave, George} and {Eileen, Barb, Jill} > >>should I write out > >> > >> Adam, Bob, Chuck, Dave, Edgar, Fred, George, Hank, Igor, Jack > >> ^^^ ^^^^ ^^^^^^ > if you insist it matches some vague unspecified notion of "same size". > It may seem unintuitive, > >> > >> ^^^^ ^^^^^^ ^^^^ > >> > >>?? How do I know what the missing elements are? > >> > >>The one-to-one correspondence idea is nice because it works for any > >>two sets. The idea you are looking at only works if one set > >>is a subset of the other. > > > Yes, to set up the paradox we need to compare two sets for which > > there is a 1:1C and one is a subset of the other. It isn't a question of > > what works. It's a question of how the paradox is to be resolved. > > > Thanks, Six Letters > > There is no need to resolve the paradox. There exists a > one-to-correspondence between the natural numbers and the > perfect squares. The perfect squares are also a proper > subset of the natural numbers. This is not a contradiction. > > Stephen There is, in fact, a definition of infiniteness of sets (by Dedekind) which REQUIRES the bijection of the set with a proper subset of itself. http://en.wikipedia.org/wiki/Dedekind_infinite
From: Ross A. Finlayson on 26 Nov 2006 22:03 Bob Kolker wrote: > Six Letters wrote: > > > > > I want to suggest there are only two sensible ways to resolve the > > paradox: > > > > 1) So- called denumerable sets may be of different size. > > Same size (i.e. same cardinality) means there is a one to one > correspondence between the sets. That is the definition of "same size". > > > > > > 2) It makes no sense to compare infinite sets for size, neither to say one > > is bigger than the other, nor to say one is the same size as another. The > > infinite is just infinite. > > False. Where a correspondence can be established it makes perfectly good > sense. > > > > > > My line of thought is that the 1:1C is a sacred cow. That there is > > no extension from the finite case. > > Dead wrong. You have been wrong for over a hundred years. > > Bob Kolker The density of infinite sets, within each other or supersets, has definite meaning and certainly corresponds to some notion of relative size. Half the integers are even. That's no longer disputed here although you can readily find in years past times when those defending cardinality as the only definition of a notion of size of sets were adamant that was not true. So, it's true, they were wrong. A proper subset is smaller than the set. The size of the proper subset is less. Remove elements from a non-empty set, there are less. Consider the notion of some probability distribution over the naturals of which you know not its form, except each integer has non-zero probability in selection, that selects a (natural) integer. In a wager for something you value, would you be willing to accept 10:1 odds in your favor that it was a multiple of 12? 1200? Is the density of a subset of the natural integers in the naturals integers a meaningful comparison of its size compared to the density of other subsets of the natural integers? It certainly is. Would you think a random integer is more likely to be a multiple of two, or a multiple of two billion? I think you would agree that on the average, or on the whole, it's a billion times more likely that a random integer is a multiple of two than that it is as well a multiple of two billion. While Galileo illustrated that the squares in progression could be mapped 1-1 to the naturals, that they were equivalent, that doesn't change that only half the integers are even. The density of a set of the natural integers, which is a very particular set of numbers where all the number-theoretic statements about them apply, is a number between zero and one, and allows much more precision, in the comparison of sizes of subsets of the natural integers within the natural integers, for a wider variety of infinite sets than does cardinality, which says only nothing about their contrast. I would continue, but the points are that: a) a proper subset size relation exists, and b) relative density is relative size, where the set of natural integers has one. There is no universe in ZF. Quantification and identity are too big for ZF. Sum the differentials, it's called integration. How many copies of the reals does it take to biject to N^N? The universe is infinite, then infinite sets are equivalent, because the universe is an example that infinite set and powerset, itself, are equivalent. Ross
From: Eckard Blumschein on 27 Nov 2006 04:05
On 11/24/2006 11:25 PM, Virgil wrote: > In article <45675C18.5090705(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 11/24/2006 12:35 PM, Six wrote: >> > GALILEO'S PARADOX >> >> >> I already wrote elsewhere that Galilei eventually found out: >> The relations smaller, equally large, and larger do not hold for >> infinite quantities. > > Depends on how one defines them. Cantor's definition works quite well > for infinite sets, at least in ZFC and NBG. I wonder if the relations smaller, equally large, and larger require any comment or definition. I also got not aware of a redefinition by Cantor. >> Dedekind and Cantor intended to make possible the impossible and just >> enchant irrationals into numbers of the same kind as rational numbers. >> If this would be possible, then there were indeed more real than >> rational numbers. > > It was and there are. Amen. |